Worked Example • Square of 3 vs. Dozen

LCM of 9 and 12 (Equals 36)

The Least Common Multiple of 9 and 12 is 36. Both numbers are divisible by 3 (GCF = 3), so LCM(9, 12) = (9 × 12) ÷ 3 = 36.

LCM of 9 and 12 Result

LCM(9, 12) = 36

Greatest Common Factor: GCF(9, 12) = 3

Check the answer (every division leaves a remainder of 0):

36 ÷ 9 = 436 ÷ 12 = 3

Show the Working (Step-by-Step)

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Step 1: Break each number down into its prime factors.

Write each input as a product of prime numbers, then group matching primes with exponents.

9 = 3 × 3

= 3²

12 = 2 × 2 × 3

= 2² × 3

Step 2: For each prime, pick the highest exponent that appears in any single number.

Prime	In 9	In 12	Highest Power
2	—	2²	2² = 4
3	3²	3¹	3² = 9

Step 3: Multiply those highest prime powers together to get the LCM.

LCM = 2² × 3² = 4 × 9 = 36

If you are copying the steps onto a homework worksheet, here is how each method reaches LCM(9, 12) = 36:

Multiply the highest power of each prime: 2² × 3² = 36.

The Greatest Common Factor of 9 and 12 is 3.

36 ÷ 9 = 4 (remainder 0 ✓)

36 ÷ 12 = 3 (remainder 0 ✓)

In yardage and foot measurements, a 36-inch yardstick can be divided cleanly into 9-inch segments (4 segments) or 12-inch feet (3 feet).

Don't multiply 9 × 12 = 108. Stepping by 12s (12, 24, 36) reaches a multiple of 9 on the very third step.